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32     ON THE EFFECT OF  THE  INTERNAL  FRICTION  OF  FLUIDS
the suffix a denoting that r is supposed to have the value a in the general expressions for Pr and 2V
The expression for F may be greatly simplified, without employing the solution of equations (27), (28), by combining these equations in their original state with the equations of condition (30). We have, in the first place, from (26)
.__
rsmfldr
Now the equations (30) make kuown the values of ^ and dty/dr, and of their differential coefficients of all orders with respect to 6, •when r = a. When the expressions for R and @ are substituted in (46), the result will contain only one term in which the differentiation with respect to r rises to the second • order. But we get from (21), (27), (28) •

_                  .
dr2          r2   d0 sin 6 d0    V dt '
and the second of equations (30) gives the value for r = a of the first term in the right-hand member of the equation just written. We obtain from (48) and (30)
(dE\        si ~
\rdBja         a   dt     \rja'
\ drja        fju a sin 6 \ dt ) a ' Substituting in (47), and writing pp for p, we get
F = 2™ f '{ - apa cos 0 + p (^] } sin 6d6. J o l                       \ at /a)
With respect to the first term in this expression, we get by integration by parts
The first term vanishes at the limits.   Substituting in the second